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 A287650 Number of doubly symmetric diagonal Latin squares of order 2n with constant first row. 4
 0, 2, 0, 12288 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A doubly symmetric square has symmetries in both the horizontal and vertical planes. One plane symmetry requires one-to-one correspondence between values of elements a[i][j] and a[N-i][j] in vertical plane and between values of elements a[i][j] and a[i][N-j] in horizontal plane for all i and j values (numbering if indexes from 1). - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017 It seems that doubly symmetric diagonal Latin squares exists only for orders N == 0 (mod 4). - Eduard I. Vatutin, Oct 18 2017 LINKS E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, value a(4) is wrong (in Russian) E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian) E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian) E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79. E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 17-19 (in Russian) E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 30-36 (in Russian) FORMULA a(n) = A292517(n)/n!. EXAMPLE Doubly symmetric diagonal Latin square example: 0 1 2 3 4 5 6 7 3 2 7 6 1 0 5 4 2 3 1 0 7 6 4 5 6 7 5 4 3 2 0 1 7 6 3 2 5 4 1 0 4 5 0 1 6 7 2 3 5 4 6 7 0 1 3 2 1 0 4 5 2 3 7 6 Reversion of all rows is equivalent to the exchange of elements 0 and 7, 1 and 6, 2 and 5, 3 and 4; hence, this square is horizontally symmetric. Reversion of all columns is equivalent to the exchange of elements 0 and 1, 2 and 4, 3 and 5, 6 and 7; hence, the square is also vertically symmetric. CROSSREFS Cf. A003191, A287649, A292517. Sequence in context: A265594 A269904 A090446 * A270830 A270829 A104157 Adjacent sequences:  A287647 A287648 A287649 * A287651 A287652 A287653 KEYWORD nonn,more,bref,changed AUTHOR Eduard I. Vatutin, May 29 2017 EXTENSIONS a(4) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017 STATUS approved

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